Error analysis of a BDF2-FEM discretization for flow in deformable porous media

In this work, we consider a linearized model for fluid flow through a deformable porous elastic solid derived using mixture theory. The model accounts for small but non-negligible solid velocity relative to the fluid, resulting in a coupled system of equations. We focus on the error analysis and implementation of two fully discrete schemes: one using the first-order backward Euler method for time discretization and the other employing the second-order BDF2 scheme. We derive error estimates for both procedures and describe numerical experiments on the corresponding non-dimensional system.